Subdimensional single-carrier modulation

ABSTRACT

A method for modulating a sequence of data symbols such that the transmit signal exhibits spectral redundancy. Null symbols are inserted in the sequence of data symbols such that a specified pattern of K data symbols and N−K null symbols is formed in every period of N symbols in the modulated sequence, N and K being positive integers and K being smaller than N.

CROSS-REFERENCE TO RELATED APPLICATION(S)

The present application is a continuation of U.S. patent applicationSer. No. 09/652,721, filed Aug. 31, 2000, which claims priority on thebasis of the provisional application Ser. No. 60/151,680, entitled“Subdimensional Single-Carrier Modulation,” filed on Aug. 31, 1999, thecontents of which are herein incorporated by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to methods and systems forproviding spectral redundancy while modulating information to betransmitted with a single carrier signal.

2. Description of Related Art

Narrowband interference and deep spectral notches in the transmissionchannel of single-carrier modulation (SCM) systems are commonimpairments in high-speed digital transmission over twisted-pairsubscriber lines, home phone-line networks, upstream transmission inCATV cable systems, and wireless communication systems. For example,these impairments can occur in Very-High Speed Digital Subscriber Line(VDSL) systems, which are currently in a definition phase. In a VDSLsystem, signals will be transmitted over twisted-pair subscriber linesin the frequency band from a few 100 kHz up to 20 MHz. Cable attenuationand crosstalk from other pairs in the same cable binder are the mainimpairments. Deep spectral notches in the channel transfer function maybe caused by bridged taps. In addition, spectral notches may beintentionally introduced at the transmitter to prevent radiation fromthe cable into certain RF bands (such as amateur radio bands).Narrowband RF interference must be suppressed by notching thecorresponding bands at the receiver. The characteristics of thenarrowband impairments are often a priori unknown at the receiver andmay change over time.

Adaptive decision-feedback equalization (DFE) is conventionally used todeal with these impairments. Good performance is achieved ifinterference levels and the depth of spectral notches are moderate, orthe width of impaired spectral regions is small compared to the Nyquistbandwidth. However, if the impairments are more severe, DFE requireslong feedforward filter and feedback filters, and system performance, asmeasured by the mean-square error, is generally degraded. Moreover, thecoefficients of the feedback filter tend to become large and unendingerror propagation can occur.

This error propagation problem can be avoided by performing the feedbackfiltering operation together with modulo signal reductions in thetransmitter, instead of the receiver. This so-called “preceding”technique allows obtaining a substantially intersymbol interference(ISI) free signal at the output of the feedforward equalizer in thereceiver. Precoding also enables the use of trellis-coded modulation(TCM) or similar signal-space coding techniques on ISI channels.However, the capabilities of DFE and preceding are limited. If theimpaired spectral regions are too wide, a SCM system must avoid theseregions. Moreover, preceding requires sending the feedback filtercoefficients from the receiver to the transmitter.

Thus, there is a need for SCM systems which can deliver practicallyISI-free signals in spite of narrowband interference and deep spectralnotches in the transmission channel, and which do not have the problemsassociated with DFE and preceding.

SUMMARY OF THE INVENTION

The present invention provides a method and a system for modulating asequence of data symbols such that the modulated sequence has spectralredundancy. Null symbols are inserted in the sequence of data symbolssuch that a specified pattern of K data symbols and N−K null symbols isformed in every period of N symbols in the modulated sequence, N and Kbeing positive integers and K being smaller than N. The positions of theK data symbols within every period of N symbols are defined by an indexset.

The present invention also provides a method for processing a receivesequence. The receive sequence corresponds to a transmit signal having aspecified pattern of K data symbols and N−K null symbols within everyperiod of N symbols. N and K are positive integers and K is smaller thanN. The positions of the K data symbols within every period of N symbolsare defined by an index set. The receive sequence is equalized with atime-varying equalizer having K sets of coefficients. The K sets ofcoefficients are used periodically in accordance with the index set, toproduce an equalized receive sequence substantially free of intersymbolinterference.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects and advantages of the presentinvention will be more fully understood when considered with respect tothe following detailed description, appended claims and accompanyingdrawings, wherein:

FIG. 1 illustrates an exemplary complex-baseband model of a SD-SCMsystem.

FIG. 2A illustrates examples of maximally suppressed bands for the caseN=7, K=6.

FIG. 2B illustrates examples of maximally suppressed bands for the caseN=4, K=2, Ψ={0, 1}.

FIG. 3 illustrates the operation of a time-varying equalizer of lengthon the received signal sequence {x_(n)} for the example N=8, K=3, Ψ={0,1, 3}.

FIG. 4 illustrates an embodiment of the time-varying equalizer 114 (FIG.1).

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a method and a system for reliablecommunication in the presence of narrowband interference and deepspectral notches in the transmission channel. Unlike conventionalmethods, the method of the present invention does not have the problemsassociated with DFE and precoding (discussed in the Background section).

The method of the present invention is hereinafter referred to as“Subdimensional Single-Carrier Modulation” (SD-SCM) technique, and thecorresponding system is hereinafter referred to as SD-SCM system.

Generally, subdimensional modulation means that signals are transmittedin a subspace of the signal space supported by a given channel. Achannel with single-sided bandwidth of W hertz admits a signal space of2W real signal dimensions per second. This maximum rate of dimensionsper second can be achieved in many ways. Sending W=1/T complex symbolsper second by quadrature-modulated SCM (with ideal “brick wall” pulseshaping filter) is one such example. Restricting modulation to subspacesof the supported signal space can also be accomplished in many ways. Themethod of the present invention provides one way of restricting themodulation to a K-dimensional subspace of the supported N-dimensionalsignal space. The important aspect of the present invention is to sendsignals with spectral support over the full bandwidth of the channel andachieve spectral redundancy by modulation constraints. Spectralredundancy can then be exploited in the receiver to recover thetransmitted symbols from the spectral regions that have goodtransmission characteristics.

In a SD-SCM system, the transmitter inserts null symbols periodically inthe sequence of data symbols. In the simplest case, one null symbol isinserted after every N−1 data symbols. In the general case, in everyperiod of N symbols, a specified pattern of K data symbols and N−K nullsymbols is transmitted. The receiver includes a time-varying linearfeedforward equalizer with K sets of coefficients. Each of the K sets ofcoefficients is periodically used to equalize the K data symbols inevery N-symbol period. It will be shown that ISI-free symboltransmission can be achieved if the combined bandwidth of severelynotched or disturbed spectral regions does not exceed (N−K)/N×(1/T),where 1/T denotes the Nyquist bandwidth.

The SD-SCM technique does not preclude adding a decision feedback filterin the receiver, or performing preceding in the transmitter. However,DFE or preceding are not essentially required in a SD-SCM system.

The coefficients of the time-varying feedforward equalizer can beadjusted adaptively using the least-mean-squares (LMS) algorithm. Therequired coordination between the transmitter and the receiver of aSD-SCM system is minimal. Blind equalization is also possible.

FIG. 1 illustrates an exemplary complex-baseband model of a SD-SCMsystem. The symbol response of the transmit filter and the impulseresponses of the channel and the receive filter are denoted by h_(T)(t),g(t), and g_(R)(t), respectively. Generally, these functions are complexvalued. The corresponding Fourier transforms are H_(T)(f), G(f), andG_(R)(f). Complex symbol transmission at modulation rate 1/T is assumed.

Referring to FIG. 1, a modulator 102 accepts information bits at itsinputs and maps them to data symbols. The modulator 102 also insertsnull symbols in the sequence of the data symbols such that a specifiedpattern of K data symbols and N−K null symbols is formed in every periodof N symbols. N and K are positive integers and K is smaller than N. Thepositions of the K data symbols within every period of N symbols aredefined by an index set Ψ which is a subset of the set {0, 1, 2, . . . ,N−1}:

Ψ{k ₀ ,k ₁ , . . . k _(K−1);0≧k _(i) ≧N−1,k _(i) ≠k _(j) for i≠j}  (1)

It is assumed that Ψ is free of subperiods; otherwise, an N-symbolperiod can be shortened to contain only one subperiod withcorrespondingly smaller values of N and K. Without loss of essentialgenerality, one may require that k₀=0.

The symbol sequence {a_(n)=a_(iN+k)} at the output 103 of the modulator102 is of the form

${a_{{iN} + k} = \begin{Bmatrix}{{data}\mspace{14mu} {symbol}} & {{{if}\mspace{14mu} k} \in \Psi} \\0 & {{{if}\mspace{14mu} k} \notin \Psi}\end{Bmatrix}},{i \in Z},{0 \leq k \leq {N - 1}},$

where Z denotes the set of integers.

The symbol sequence {a_(n)}is converted to an analog transmit signal byan digital-to-analog converter 104 operating at a clock times nT and apulse shaping transmit filter 106. The transmit filter may be a raisedcosine filter. The communication channel 108 generally distorts thetransmitted signal and adds noise to it. For example, in the case ofmetallic twisted-pair cables, increasing attenuation with frequency andreflections of unused branching sections of the communication line(bridged taps) cause signal distortion. At higher frequencies, bridgedtaps can lead to deep notches in the spectral of the received signal.Noise may be due to near-end (NEXT) and/or far-end crosstalk (FEXT) fromother cables in the same cable binder, ingress of narrowband radiointerference, or disturbances from electric motors and householdappliances. In FIG. 1, N(f) denotes the power spectral density ofadditive noise.

The SD-SCM system may also include in the transmit filter a notch filterto produce notches in spectrum of the transmitted signal atpredetermined frequency bands, the notches having an aggregate bandwidthof less than or equal to ((N−K)/N)×(1/T), where 1/T is the Nyquistbandwidth. For example, in VDSL systems such notching may be necessaryto prevent egress radiation from a cable into amateur-radio bands.

At the input of the receive filter 110, a continuous time (analog)signal x(t) is received. A sampler 112, i.e., an analog-to-digitalconverter, samples x(t) at clock times nT+τ and produces a signalsequence {x_(n)}. In this case, the time spacing between the sampledsignals is equal to T. In other embodiments of the invention thereceived signal may be sampled at higher rates.

A time-varying equalizer 114 having K sets of coefficients operates onthe T-spaced sequence {x_(n)} and produces an output sequence {y_(n)}.For every N-symbol period in {x_(n)}, the equalizer 114 outputs Kequalized signals. In the absence of noise, these K signals willpractically be equal to the K data symbols in the original transmitsignal, i.e., y_(iN+k)≈a_(iN+k), k∈Ψ, provided the index set Ψ is chosensuch that zero-ISI signals can be delivered in the presence of stronglyimpaired spectral regions of aggregate width of up to (N-K)/(NT) hertz.This can be achieved when the index set Ψ is a “proper set”. Thedefinition of a proper set is given in terms of the K×K matrix A_(K×K)(

, Ψ) defined in Equation (18). If A_(K×K)(

, Ψ) is non-singular for all possible sets L, then Ψ is called a properset.

One sufficient condition for a proper set Ψ is that N is a prime number.In this case, Ψ can be of any K-ary subset of {0, 1, 2 . . . , N−1}.

Another sufficient condition for a proper set Ψ is that the index set Ψis equal to {k_(i)=k₀iΔ mod N, i=0, 1, 2, . . . , K−1}, where 0≦k₀≦N−1and Δ is an integer and such that the greatest common divisor of Δ and Nis 1. In this case, the index set Ψ is called “proper sequential”. Aspecial set of this type is obtained when K is equal to N−1. The singlenull symbol may be inserted anywhere in an N-symbol period.

Examples of proper sets Ψ with k₀₌₀ (=Ψ₀) are:

N≧2,1≦K≦N−1, Δ=1: Ψ₀={0, 1, 2, . . . , K−1}

N=8, K=6, ΔΔ=3: Ψ₀={0, 3, 6, 1, 4, 7}={0, 1, 2, 3, 4, 5, 6, 7} (elementsreordered)

N=11, K=8: Ψ₀{0, 2, 3, 4, 5, 6, 7, 8, 9} (N prime).

It is noted that there exists also proper sets of Ψ, which do notsatisfy one of the above two sufficient conditions, for example:

N=8, K=3: Ψ={0, 1, 3}

N=10, K=7: Ψ={0, 1, 2, 3, 4, 5, 6, 7, 8}

Mathematically, usage of a pattern of K data symbols out of N symbolscan be interpreted as K-dimensional subspace modulation in anN-dimensional signal-vector space. The modulation subspace can berotated by multiplying N-dimensional signal vectors by an arbitrary N×Nunitary matrix.

Any pattern that can be expressed as a rotation of a given pattern isconsidered equivalent to the given pattern for the purpose of the methodof the present invention.

When the index set Ψ is a proper set, the time-varying equalizer 114 cansuppress strongly impaired spectral regions of aggregate width of up to(N−K)/(NT) hertz while delivering practically zero-ISI signals. Thiswill be shown mathematically later.

The impaired spectral regions do not have to be in connected bands. Aslong as their aggregate width is less than or equal to (N−K)/(NT) hertzin the Nyquist band f∈[0, 1/T), the time-varying equalizer 114 canproduce practically ISI-free signals. Examples illustrating zero-ISIachievement with maximally suppressed bands are shown in FIG. 2A andFIG. 2B.

FIG. 2A illustrates examples of maximally suppressed bands for the caseN=7, K=6. In this case, there are 6 data symbols and a single nullsymbol in every period of 7 symbols. The maximum aggregate width ofstrongly impaired spectral regions is (N−K)/(NT)= 1/7T hertz. In part(a) of FIG. 2A, the suppressed spectral region in is one sub-band ofwidth 1/7T. In part (b) of FIG. 2A, the suppressed spectral regions aretwo non-contiguous regions of unequal widths, the aggregate width ofwhich is equal to 1/7T. In part (c) of FIG. 2A, the suppressed spectralregions are three non-contiguous regions in the Nyquist band f∈[0, 1/T),the aggregate width of which is equal to 1/7T.

FIG. 2B illustrates examples of maximally suppressed bands for the caseN=4, K=2, Ψ={0 μl}. In this case, there are 2 data symbols and two nullsymbols in every period of 4 symbols. As indicated by Ψ, the datasymbols are at positions 0 and 1 in every 4-symbol period. The maximumaggregate width of strongly impaired spectral regions is (N−K)/(NT)=½Thertz. In part (a) of FIG. 2B, the suppressed spectral region in is oneband of width ½T. In part (b) of FIG. 2B, the suppressed spectralregions are three non-contiguous regions of unequal widths in theNyquist band f∈[0, 1/T), the aggregate width of which is equal to ½T. Inpart (c) of FIG. 2B, the suppressed spectral regions are sevennon-contiguous regions in the Nyquist band f∈[0, 1/T), the aggregatewidth of which is equal to ½T.

The following describes the operation of the time-varying equalizer 114(FIG. 1) on the T-spaced sequence {x_(n)} with K sets of coefficients.For clarity, the description will be for a specific example. The exampleis the case N=8, K=3, Ψ={0, 1, 3} (a proper set, as stated above). Inone embodiment, the time-varying equalizer 114 (FIG. 1) is a finiteimpulse response filter of sufficient length M (generally greater thanN). For example, M=10. The time-varying equalizer has M−1=9 delayelements. The time-varying equalizer operates on the T-spaced sequence{x_(n)} with K=3 sets of coefficients as follows. The symbols of thesequence {x_(n)} are shifted into the equalizer 114 one symbol at atime. Except for the current symbol, the symbols are stored in the delayelements of the equalizer. The input symbols are multiplied by thecoefficients of the currently used set of coefficients. The results arecombined to produce an output symbol. The equalizer has 3 sets ofcoefficients. Each of the 3 sets has 10 coefficients and is usedperiodically.

The equalizer uses the first set of coefficients, corresponding to k=0,i.e., the first element of Ψ, to operate on the following 10 inputsymbols and outputs symbol y_(iN):

{X_((i−1)N−1)X_((i−1)N)X_(iN−7)X_(iN−6)X_(iN−5)X_(iN−4)X_(iN−3)X_(iN−2)X_(iN−1)X_(iN)}

The equalizer uses the second set of coefficients, corresponding to k=1,i.e., the second element of Ψ, to operate on the following 10 inputsymbols and outputs symbol y_(iN+1):

{X_((i−1))X_(iN−7)X_(iN−6)X_(iN−5)X_(iN−4)X_(iN−3)X_(iN−2)X_(iN−1)X_(iN)X_(iN+1)}

The equalizer uses the third set of coefficients, corresponding to k=3,i.e., the third element of Ψ, to operate on the following 10 inputsymbols and outputs symbol y_(iN+3):

{X_(iN−6)X_(iN−5)X_(iN−4)X_(iN−3)X_(iN−2)X_(iN−1)X_(iN)X_(iN+1)X_(iN+2)X_(iN+3)}

FIG. 3 illustrates the operation of the time-varying equalizer on theinput symbol sequence {x_(n)}. After the first set of coefficients isused to operate on the 10 input symbols with x_(iN) being the mostcurrent input symbol, a new input symbol x_(iN+1) is shifted into theequalizer. The second set of coefficients is then used to operate on thegroup of 10 input symbols with x_(iN+1) being the most current inputsymbol. Then, x_(iN+2) is shifted in, but the equalizer does not operateon this group of 10 symbols. Then, x_(iN+3) is shifted in, and the thirdset of coefficients is used to operate on the group of 10 symbols withx_(iN+3) being the most current input symbol. The elements in the indexset Ψ={0, 1, 3} determine at what positions in every N-symbol period ofthe input sequence the K sets of coefficients are used. In the aboveexample, they are used at positions 0, 1, 3. After the third set ofcoefficients is used, the first set of coefficients is used again whenx_((i+1)N) is shifted in as the most current input symbol. And the cyclecontinues.

FIG. 4 illustrates an embodiment of the time-varying equalizer 114 (FIG.1). The K sets of M coefficients {c₀, c₁, . . . , c_(M−1)} are stored inM circular buffers 402, one circular buffer for each of the Mcoefficient positions of the equalizer. During every N-symbol period,the K coefficient values in every circular buffer are cyclically appliedto generate K equalizer output signals. Additional control may benecessary to ensure that the sequence of obtained K equalized signalscorresponds to the sequence of data symbols defined by Ψ withoutduplication or omission of symbols.

This is just one exemplary implementation of the time-varying equalizer.Other architectures are possible.

The following discussion will show that when the index set Ψ is a properset, the time-varying equalizer 114 can suppress strongly impairedspectral regions of aggregate width of up to (N−K)/(NT) hertz whiledelivering practically zero-ISI signals.

The continuous-time signal at the receive-filter output isx(t)=Σ_(n)a_(n)h(t−nT)+v(t). Sampling at times nT+τ yields

$\begin{matrix}{{x_{n} = {{x\left( {{nT} + \tau} \right)} = {{\sum\limits_{n}\; {h_{}a_{n - }}} + v_{n}}}},} & (2) \\{where} & \; \\\begin{matrix}{h_{} = {h\left( {{\; T} + \tau} \right)}} \\{{= {\int{H(f)^{{j2}\; \pi \; {f{({{\; T} + \tau})}}}{f}}}};} \\{{{H(f)} = {{H_{T}(f)}{G(f)}{G_{R}(f)}}},}\end{matrix} & (3) \\{{{E\left\{ {{\overset{\_}{v}}_{n}v_{n + }} \right\}} = {\int{{V(f)}^{{j2}\; \pi \; f\; \; T}{f}}}};{{V(f)} = {{N(f)}{{{G_{R}(f)}}^{2}.}}}} & (4)\end{matrix}$

Hereinafter, 1/T-periodic spectral functions are denoted with a tildeand considered in the Nyquist band f∈[0, 1/T), rather than [−½T, +½T).The 1/T-periodic spectral symbol response {tilde over (H)}(f) and thepower spectral density of the noise {tilde over (V)}(f) are:

$\begin{matrix}{{{\overset{\sim}{H}(f)} = {{\sum\limits_{}\; {h_{}^{{- {j2}}\; \pi \; f\; \; T}}} = {\frac{1}{T}{\sum\limits_{i}\; {{H\left( {f + {i/T}} \right)}^{{j2}\; \pi \; {({f + {i/T}})}\tau}}}}}},{f \in \left\lbrack {0,{1/T}} \right)},} & (5) \\{{{\overset{\sim}{V}(f)} = {{\sum\limits_{}\; {E\left\{ {{\overset{\_}{v}}_{n}v_{n + }} \right\} ^{{- {j2}}\; \pi \; f\; \; T}}} = {\frac{1}{T}{\sum\limits_{i}\; {V\left( {f + {i/T}} \right)}}}}},{f \in {\left\lbrack {0,{1/T}} \right).}}} & (6)\end{matrix}$

The equalizer is a time-varying FIR filter operating on the T-spacedsequence {x_(n)} with K sets of coefficients {d_(l) ^((k))}, k∈Ψ. The Ksymbol responses at the K equalizer outputs are:

$\begin{matrix}{{{\forall{k \in {\text{Ψ:}s_{}^{(k)}}}} = {{\sum\limits_{^{\prime}}\; {d_{^{\prime}}^{(k)}h_{ - ^{\prime}}}} = {{\int_{0}^{1/T}{{\overset{\sim}{H}(f)}{{\overset{\sim}{D}}^{(k)}(f)}\ ^{{j2}\; \pi \; f\; \; T}{f}}} = {\int_{0}^{1/T}{{{\overset{\sim}{S}}^{(k)}(f)}^{{j2}\; \pi \; f\; \; T}{f}}}}}},} & (7) \\{where} & \; \\{{{\forall{k \in {\text{Ψ:}{{\overset{\sim}{D}}^{(k)}(f)}}}} = {\sum\limits_{}\; {d_{}^{(k)}^{{- {j2}}\; \pi \; f\; \; T}}}},{f \in \left\lbrack {0,{1/T}} \right)},} & (8) \\{{{\forall{k \in {\text{Ψ:}{{\overset{\sim}{S}}^{(k)}(f)}}}} = {{\sum\limits_{}\; {s_{}^{(k)}^{{- {j2}}\; \pi \; f\; \; T}}} = {{\overset{\sim}{H}(f)}{{\overset{\sim}{D}}^{(k)}(f)}}}},{f \in {\left\lbrack {0,{1/T}} \right).}}} & (9)\end{matrix}$

In the absence of noise, the K outputs of the equalizer in the i-thN-symbol period are given by

$\begin{matrix}{{\forall{k \in {\text{Ψ:}y_{{iN} + k}}}} = {{\sum\limits_{}\; {d_{}^{(k)}x_{{iN} + k - }}} = {\sum\limits_{{k - {\mspace{11mu} {mod}\mspace{11mu} N}} \in K}\; {s_{}^{(k)}{a_{{iN} + k - }.}}}}} & (10)\end{matrix}$

The second summation in (10) accounts for data symbols only, i.e., nullsymbols are excluded.

The following discussion addresses the conditions for zero intersymbolinterference (ISI) at the equalizer output. In a subdimensional SCMsystem with a rate of K/N, ISI-free transmission can be accomplishedwithin a minimal one-sided bandwidth of K/NT Hz. Available choices forspectral suppression in the Nyquist bandwidth 1/T Hz will be examined.Absence of noise is assumed.

Zero ISI requires y_(iN+k)=a_(iN+k), k∈Ψ. From Equation (10), thetime-domain conditions for zero-ISI are:

$\begin{matrix}{{\forall{k \in {\text{Ψ:}s_{}^{(k)}}}} = \left\{ {\begin{matrix}\delta_{} & {{{if}\mspace{14mu} \left( {k - {\mspace{14mu} {mod}\mspace{14mu} N}} \right)} \in \Psi} \\u_{}^{(k)} & {{{if}\mspace{14mu} \left( {k - {\mspace{14mu} {mod}\mspace{14mu} N}} \right)} \notin \Psi}\end{matrix}.} \right.} & (11)\end{matrix}$

In Equation (11), δ_(l) is the Kronecker indicator function, defined asbeing equal to zero for all l≠0 and equal to 1 for l=0. The values ofu_(l) ^((k)) can be arbitrary.

The following shows the values of the symbol responses s_(l) ^((k)) forthe example where N=4, K=3, Ψ={0, 1, 2}.

$\begin{matrix}\text{:} & \ldots & {- 7} & {- 6} & {- 5} & {- 4} & {- 3} & {- 2} & {- 1} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & \ldots \\{s_{}^{(0)}:} & \ldots & u & 0 & 0 & 0 & u_{- 3}^{(0)} & 0 & 0 & 1 & u_{1}^{(0)} & 0 & 0 & 0 & u & 0 & 0 & 0 & u & \ldots \\{s_{}^{(1)}:} & \ldots & 0 & u & 0 & 0 & 0 & u_{- 2}^{(1)} & 0 & 1 & 0 & u_{2}^{(1)} & 0 & 0 & 0 & u & 0 & 0 & 0 & \ldots \\{s_{}^{(2)}:} & \ldots & 0 & 0 & u & 0 & 0 & 0 & u_{- 1}^{(2)} & 1 & 0 & 0 & u_{3}^{(2)} & 0 & 0 & 0 & u & 0 & 0 & \ldots\end{matrix}$

The ambiguity of the symbol responses in the locations of the u_(l)^((k))'s implies spectral redundancy. The nature of this redundancy willbecome apparent by expressing the fixed part of Equation (11), i.e., theupper part in Equation (11), in frequency-domain terms.

It is appropriate to divide the Nyquist band [0, 1/T) into N subbands[l/NT, (l+1)/NT), l=0, 1, . . . N−1, and denote the spectral symbolresponses in these subbands of equal width 1/NT by:

∀k∈Ψ,l=0,1, . . . N−1: S _(l) ^((k))(f′)={tilde over (S)}^((k))(f′+l/NT),f′∈[0,1/NT).  (12)

With l=k−k′+iN, k′∈Ψ, the fixed part of Equation (11) is written as:

$\begin{matrix}{{\forall{k \in {\text{Ψ,}\mspace{11mu} k^{\prime}} \in {\text{Ψ:}{\sum\limits_{i}\; {s_{k - k^{\prime} + {iN}}^{(k)}^{{j2}\; \pi \; {f{({k - k^{\prime} + {iN}})}}T}}}}}} = {\delta \; {\delta_{k - k^{\prime}}.}}} & (13)\end{matrix}$

Substitution of

$\begin{matrix}{\; {s_{k - k^{\prime} + {iN}}^{(k)} = {{T{\int_{0}^{1/T}{{{\overset{\sim}{S}}^{(k)}(f)}^{{j2}\; \pi \; {\overset{̑}{f}{({k - k^{\prime} + {iN}})}}T}\ {f}}}} = {T{\sum\limits_{ = 0}^{N - 1}\; {\int_{0}^{1/{NT}}{{S^{(k)}\left( f^{\prime} \right)}\ ^{{j2}\; \pi \; {({f^{\prime} + {/{NT}}})}{({k - k^{\prime} + {iN}})}T}{f^{\prime}}}}}}}}} & (14)\end{matrix}$

into Equation (13) yields the zero-ISI conditions in thefrequency-domain form:

$\begin{matrix}{{{\forall{k \in {\text{Ψ,}\mspace{11mu} k^{\prime}} \in {\text{Ψ:}{\sum\limits_{ = 0}^{N - 1}\; {{s_{}^{(k)}\left( f^{\prime} \right)}^{{j2}\; \pi \; {{{({k - k^{\prime}})}}/N}}}}}}} = {N\mspace{11mu} \delta_{k - k^{\prime}}}},{f^{\prime} \in {\left\lbrack {0,{1/{NT}}} \right).}}} & (15)\end{matrix}$

The trivial solutions of (15) are S_(l) ^((k))(f′)=1 for all l={0, 1, 2,. . . N−1}. This corresponds to the well-known Nyquist criterion {tildeover (S)}^((k))(f)={tilde over (H)}(f){tilde over (D)}^((k))(f)=1 forzero-ISI in unconstrained successions of modulation symbols (K=N). Thissolution is achievable with {tilde over (D)}(k)(f)={tilde over(H)}⁻¹(f), if {tilde over (H)}(f) exhibits spectral support in theentire Nyquist band [0, 1/T). If {tilde over (H)}(f) does not have fullspectral support, or the equalizer has to suppress severe narrowbandinterference at certain frequencies, then {tilde over (S)}^((k))(f)should vanish in the affected spectral regions.

It will be shown that solutions of Equation (15) exist with S_(l)^((k))(f′)≠0, l∈

, and S_(l) ^((k))(f′)=0 (or arbitrary fixed values), l∉

, for all possible

:

={l ₀ ,l ₁ , . . . l _(K−1):0≦l _(i) ≦N−1;l _(i) ≠l _(j) ^(,i≠j}.)

In other words, {tilde over (S)}^((k))(f) can vanish at any combinationof up to N−K frequencies in the set of frequencies {f=f′+l/NT, l=0, 1,2, . . . N−1)}, for every frequency f′68 [0,1/NT). The suppression of(N−K)/NT Hz in not necessarily connected bands is illustrated in FIG. 2Aand FIG. 2B by several examples (discussed above).

Let k∈Ψ and S_(l) ^((k))(f′)=0, l∉

Then, with α_(N)=e^(j1π/N), Equation (15) can be written as a system ofK linear equations for K unknowns:

$\begin{matrix}{{{{A_{K \times K}\left( {,\Psi} \right)} \times \begin{bmatrix}{\alpha \; \alpha_{N}^{{- _{0}}k}{S_{_{0}}^{(k)}\left( f^{\prime} \right)}} \\{\alpha \; \alpha_{N}^{{- _{1}}k}{S_{_{1}}^{(k)}\left( f^{\prime} \right)}} \\\vdots \\{\alpha \; \alpha_{N}^{{- _{K - 1}}k}{S_{_{x - 1}}^{(k)}\left( f^{\prime} \right)}}\end{bmatrix}} = {N\begin{bmatrix}{\delta \; \delta_{k - k_{0}}} \\{\delta \; \delta_{k - k_{1}}} \\\vdots \\{\delta \; \delta_{k - k_{K - 1}}}\end{bmatrix}}},} & (17) \\{and} & \; \\{{A_{K \times K}\left( {,\Psi} \right)} = {\begin{bmatrix}{\alpha \; \alpha_{N}^{{- _{0}}k_{0}}} & {\alpha \; \alpha_{N}^{{- _{1}}k_{0}}} & {\alpha \; \alpha_{N}^{{- _{2}}k_{0}}} & \ldots & {\alpha \; \alpha_{N}^{{- _{K - 1}}k_{0}}} \\{\alpha \; \alpha_{N}^{{- _{0}}k_{1}}} & {\alpha \; \alpha_{N}^{{- _{1}}k_{1}}} & {\alpha \; \alpha_{N}^{{- _{2}}k_{1}}} & \ldots & {\alpha \; \alpha_{N}^{{- _{K - 1}}k_{1}}} \\{\alpha \; \alpha_{N}^{{- _{0}}k_{2}}} & {\alpha \; \alpha_{N}^{{- _{1}}k_{2}}} & {\alpha \; \alpha_{N}^{{- _{2}}k_{2}}} & \ldots & {\alpha \; \alpha_{N}^{{- _{K - 1}}k_{2}}} \\\vdots & \vdots & \vdots & \vdots & \vdots \\{\alpha \; \alpha_{N}^{{- _{0}}k_{K - 1}}} & {\alpha \; \alpha_{N}^{{- _{1}}k_{K - 1}}} & {\alpha \; \alpha_{N}^{{- _{2}}k_{K - 1}}} & \ldots & {\alpha \; \alpha_{N}^{{- _{K - 1}}k_{K - 1}}}\end{bmatrix}.}} & (18)\end{matrix}$

The K×K matrix A_(K×K)(

, Ψ) is a submatrix of the N×N transformation matrix of an N-point DFT.

For a solution of (17) to exist, A_(K×K)(

, Ψ) must be non-singular, i.e. det(A_(K×K)(

, Ψ)) #. All elements of the solution vector must then be non-zero. IfA_(K×K)(

, Ψ) is non-singular for all possible sets L, then Ψ is called a “properset”.

Only sets L₀ with l₀=0 need to be tested, because dividing the elementsin every row of A_(K×K)(

, Ψ) by the first row element does not change the matrix rank.Similarly, only sets Ψ₀ with k₀=0 need to be examined. Adding a non-zerointeger value c modulo N to all elements of a set Ψ₀ gives an equivalentset

${\Psi = {\Psi_{0} + {c\mspace{11mu} {mod}{\; \;}N\mspace{11mu} \left( {{{denoted}\mspace{14mu} {as}\mspace{14mu} \Psi}\overset{c}{\equiv}\Psi_{0}} \right)}}},$

corresponding to a situation where the N-periods are shifted by c.

Two sufficient conditions for a proper set Ψ₀ are (as statedpreviously):

1. Ψ₀={k_(i)=iΔΔ mod N, i=0, 1, 2, . . . K−1}, gcd(Δ, N)=1. In thiscase, the set Ψ₀ is called “proper sequential”)2. N is a prime number.

While certain exemplary embodiments have been described in detail andshown in the accompanying drawings, it is to be understood that suchembodiments are merely illustrative of and not restrictive on the broadinvention. It will thus be recognized that various modifications may bemade to the illustrated and other embodiments of the invention describedabove, without departing from the broad inventive scope thereof. It willbe understood, therefore, that the invention is not limited to theparticular embodiments or arrangements disclosed, but is rather intendedto cover any changes, adaptations or modifications which are within thescope and spirit of the invention as defined by the appended claims.

1. A single-carrier modulation (SCM) system, comprising: a modulator formodulating a sequence of data symbols such that the modulated sequencehas spectral redundancy, the modulator inserting null symbols in thesequence of data symbols such that a specified pattern of K data symbolsand N−K null symbols is formed in every period of N symbols in themodulated sequence, N and K being positive integers, and K being smallerthan N; and a time-varying equalizer for equalizing a receive sequencecorresponding to the modulated sequence, the time-varying equalizerhaving K sets of coefficients, each of the K sets of coefficients beingused cyclically. 2-23. (canceled)